Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3^x, find ƒ(log₃ (2 ln 3))

Verified step by step guidance

1. Recognize that the function is given as \(f(x) = 3^x\) and you need to evaluate \(f(\log_3(2 \ln 3))\).
2. Recall the definition of the function evaluation: \(f(\log_3(2 \ln 3)) = 3^{\log_3(2 \ln 3)}\).
3. Use the property of exponents and logarithms that states \(a^{\log_a(b)} = b\) for any positive \(a \neq 1\) and \(b > 0\).
4. Apply this property to simplify \$3^{\log_3(2 \ln 3)}\( directly to \)2 \ln 3$.
5. Thus, the expression \(f(\log_3(2 \ln 3))\) simplifies to \$2 \ln 3$ without further calculation.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. It models growth or decay processes and has properties such as a^(m+n) = a^m * a^n. Understanding how to manipulate and evaluate these functions is essential for solving problems involving exponents.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, defined as log_a(x), which answers the question: to what power must the base a be raised to get x? Key properties include log_a(a^x) = x and a^(log_a(x)) = x. Recognizing these inverse relationships helps simplify expressions involving logs and exponents.

Properties of Logarithms and Exponents

The properties of logarithms and exponents, such as the inverse relationship a^(log_a(x)) = x and the ability to rewrite expressions using these properties, are crucial for simplifying complex expressions. For example, evaluating f(log_3(2 ln 3)) involves applying these properties to simplify the composition of functions.